Singularities of Fitzpatrick and convex functions

Journal article

In a pseudo-Euclidean space with scalar product $S(\cdot, \cdot)$, we show that the singularities of projections on $S$-monotone sets and of the associated Fitzpatrick functions are covered by countable $c-c$ surfaces having positive normal vectors with respect to the $S$-product. By L. Zajíček [On the differentiation of convex functions in finite and infinite dimensional spaces, Czechoslovak Math. J. 29/104 (1979) 340--348], the singularities of a convex function $f$ can be covered by a countable collection of $c-c$ surfaces. We show that the normal vectors to these surfaces are restricted to the cone generated by $F-F$, where $F := {\rm cl}\,{\rm range}\,\nabla f$, the closure of the range of the gradient of $f$.

Authors: Kramkov, D; Sirbu, M

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Heldermann Verlag
• Journal: Journal of Convex Analysis
• Volume: 31
• Issue: 3
• Pages: 827–846
• Publication date: 2024-07-01
• Acceptance date: 2023-08-23
• EISSN: 2363-6394
• ISSN: 0944-6532
• Language: English
• Keywords: singularity; convexity; subdifferential; Fitzpatrick function; projection; pseudo-Euclidean space; normal vector